Mathematical biology of automobile driving: III

In a previous paper (Bull. Math. Biophysics,22, 257–262, 1960), an expression for the probability that a car jumps off a road as a function of the speed and the size of the car was derived mostly from geometric and kinematic considerations, introducing only the reaction time as a biological parameter. In subsequent papers (Bull. Math. Biophysics,29, 181–186, 187–188, 1967) a more detailed study was made of the exact shape of the tracking curve of the car which involved several biological parameters of the driver. In the present paper the results of the previous studies are combined, and a more general equation for the probability of jumping off the road is obtained. This probability, as in the earlier study, increases with the speedv, widths _o and lengthl _o of the car, and decreases with widths of the lane. However, this probability also depends on several parameters which characterize the psychobiological constitution of the driver. Unpublished experiments by Ehrlich, which corroborate the general conclusions, are briefly described.     

Mathematical biophysics of automobile driving

In a previous paper (Rashevsky, 1959,Bull. Math. Biophysics,21, 299–308) we derived an approximate expression for the maximal speed of driving in terms of the reaction time of the driver. In the present paper the possible effects of unevennesses of the pavement, of such distracting stimuli as road signs etc. on the reaction time are studied theoretically, using previous developments of the mathematical biophysics of the central nervous system. In this manner expressions are derived which determine the maximal safe speed in terms of road conditions and other distracting stimuli. Effects of those conditions on fatigue are also discussed.     

Automobile driving as psychophysical discrimination

The driver tries to keep the car in the center of the lane. If the car is too near the left edge, this causes the driver to make a “corrective” right turn. If the car is near the right edge, a “corrective” left turn is made. Therefore, a quantity which decreases with increasing distance Δ _L from the left edge may be considered as a stimulusS _R producing the reactionR _R of turning to the right. A similar situation holds for the distance Δ _R from the right edge. When the car is in the center of the lane, Δ _L = Δ _R andS _R =S _L , the stimuli are equal. We thus have here a situation analogous to the one studied by H. D. Landahl in his theory of psychophysical discrimination. In general a reactionR _R (resp.R _L ) will occur only ifR _R −R _L ≥h ^* (resp.R _L −R _R ≥h ^*) whereh ^* is a threshold. Applying Landahl’s theory to this situation, we find thath ^* determines the distance from the edge, at which a corrective turn is made. This distance is not constant, but a function of the speedv of the car. The requirement that a corrective turn should be madebeforre the car runs off the road leads to an expression for the maximum safe speed. Because of the transcendency of the equations involved, closed solutions cannot be obtained. It is, however, shown that the expression for maximum safe speed, given in a previous paper (Bull. Math. Biophysics,21, 299–308, 1959), is a rough first approximation to the expressions found now.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

Contribution to the mathematical biophysics of automobile driving

Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

Further studies on the shape of the tracking curve of an automobile and its dependence on biological parameters of the driver

Following previous studies, differential equations are established which determine the variation of the stimulus towards a corrective turn of the steering wheel and its effect on the excitation of the centers in the brain which results in the production of the corrective turn. The equations are derived under the highly oversimplified assumption that all excitation thresholds are so small that they can be neglected. Under these assumptions it is found that the tracking curve of a car is a sinusoid with negative damping, that is, with an ever increasing amplitude. Driving under these assumptions is imposible since the car will always eventually jump off the road. The possible effects of the threshold as well as stimuli towards corrective turns other than the distance from the edge of the lane are very briefly discussed. In spite of the negative results of the paper, its interest lies in the circumstance that with the complication of the model, we find that driving depends not only on the reaction times as the only “purely biological” parameter, but on three other neurobiophysical constants. In a subsequent paper (Rashevsky, 1967) it is shown how the introduction of one or more purely biological parameters of the driver makes a stable driving regime possible.     

Numerical simulations of stochastic circulatory models

Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1–22 were investigated. The models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. The reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. The Kolmogorov-Smirnov test revealed that even for 1000 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions.     

An evolution of pulse speed in arteries

In this work, treating the artery as a thick-walled cylindrical shell made of an incompressible, isotropic and elastic solid, utilizing the large deformation theory and the stress-strain relation proposed by Demiray (1976b,Trans. ASME Ser. E, J. Appl. Mech.,98, 194–197), an explicit expression for the pulse speed is obtained and the effect of lumen pressure and the axial stretch on wave speed is discussed. Numerical results indicate that the wave speed increases with lumen pressure but decreases with the axial stretch. The results of the present model are compared with our previous work (Demiray, 1988,J. Biomech. 21, 55–58) on the same subject.     

Distal Forelimb Kinematics in <Emphasis Type="Italic">Erythrocebus patas</Emphasis> and <Emphasis Type="Italic">Papio anubis</Emphasis> During Walking and Galloping

When using symmetrical gaits, terrestrial digitigrade monkeys adopt less digitigrade, i.e., more palmigrade-like, hand postures as they move with faster speeds. Accordingly, it appears that, in contrast to other mammals, digitigrady is unrelated to cursoriality in primates. However, researchers have not documented the effects of speed on distal forelimb kinematics in faster asymmetrical gaits, i.e., galloping, when ground reaction forces are typically increased owing to the decreased number of contact points during a stride, combined with higher speed. Thus, it remains possible that primates use digitigrade hand postures during these higher-speed asymmetrical gaits. We investigated 3D angles in the wrist joint and metacarpophalangeal joint of 2 habitually digitigrade terrestrial monkeys, Erythrocebus patas and Papio anubis, across a large range of walking and galloping speeds on a motorized treadmill. Nonparametric analyses reveal that angles, and therefore hand postures, are not different at the subject’s walk-gallop transition. Regression analyses show that when walking, digitigrade postures are adopted at slow speeds and more palmigrade-like postures are adopted at fast speeds. Contrary to expectations, there is little change in hand postures across galloping speeds; both subjects maintained palmigrade-like hand postures with substantial joint yield and reextension during support. These results indicate that the hands are always less digitigrade at faster speeds because the joints of the distal forelimb cannot resist the higher ground reaction forces that accompany these higher speed gaits.     

Probability distributions associated with distinct hits on targets

Some probability distributions connected with distinct hits on targets, using two different firing schemes, are developed. It is assumed that any shot has a probabilityp, not necessarily unity, of hitting the target at which it was aimed. The development uses a well-known expression for the probability that exactlyt ofN possible events occur simultaneously. Some of the formulae developed here include as special cases the probabilities derived separately and by more complicated arguments in papers by N. Rashevsky. (Bull. Math. Biophysics,17, 45–50, 1955) and A. Rapoport (Bull. Math. Biophysics,13, 133–38, 1951).     

Hunting flight speeds of five southern African raptors

The flight speeds of hunting falconry birds were determined using global positioning system data loggers. Until now, the hunting flight speed of African raptors has not been directly measured. We predicted that hunting flight speeds would differ between species and that flight dynamics, such as altitude, and bird morphology, particularly wing surface area, would influence maximum and mean flight speeds. This study considered five African raptor species, which included two long-wing species, Lanner Falcon Falco biarmicus and Peregrine Falcon F. peregrinus, one short-wing species, Black Sparrowhawk Accipiter melanoleucus, and two broad-wing species, African Hawk-eagle Aquila spilogaster and Jackal Buzzard Buteo rufofuscus. Maximum and mean hunt speeds differed significantly between the long- and short-wing species. There was no difference in acceleration or deceleration rates between these species, but this could be due to small sample sizes. There was a significant positive correlation between maximum hunt speed and maximum flight height for the long-wing species. Maximum and mean flight speeds were significantly negatively correlated with wing area for all five species in this study. However, following phylogenetic correction, no significant relationship between wing area and maximum hunt speeds was found. This study presents baseline data of hunting speeds in African raptors and further highlights the importance of inter-species variation, which can provide accuracy to flight speed models and the understanding of hunting strategies.     

Testing predictions from flight mechanical theory: a case study of Cory’s shearwater and Audouin’s gull

Predictions from flight mechanical theory concerning optimal flight speeds were tested in the field in two Mediterranean seabirds, the Cory’s shearwater Calonectris diomedea and the Audouin’s gull Larus audouinii. Both species were commuting off the coast of Isola di San Pietro, 6 km south-west of the coast of Sardinia. Heading and airspeed were obtained by vector calculation of flight tracks and measured wind. The Cory’s shearwater used a mixture of gliding and active flight. At low wind speeds the proportion of active flight was large but it decreased with increasing wind speed. The mean airspeed was 12.0 m s^–1, which is not significantly different from minimum power speed (V _mp) in active flight or the speed for best glide (V _bg) used in gliding flight. However, the shearwaters showed a significant response to wind increment/decrement, indicating that they were not flying at V _mp, which should be unaffected by head and tailwind. Furthermore, shearwaters can potentially reduce induced drag by the ground effect while flying close to the sea surface at weak winds, which leads to a reduction in characteristic flight speed. We suspect that the predictions for gliding flight are most valid for shearwaters at moderate to high wind speeds, when they should be maximising distance by using V _bg. Audouin’s gulls used active flight exclusively, with a mean airspeed of 11.3 m s^–1 that was significantly different from the predicted V _mp. Interestingly, though, the gulls did not show any significant wind response, indicating that they were flying close to their true V _mp when foraging along the coast. Received: 17 May 2000 / Received in revised form: 21 November 2000 / Accepted: 8 January 2001     

Mathematical biology of learning to drive an automobile

In learning to drive, an individual must learn to rapidly make small corrective turns to the right or to the left as the car comes too close correspondingly to the left or to the right edges of the lane. The magnitude of the corrective turn depends on the angle at which the edge is approached. Thus, the individual must learn to produce a quantitatively correct response (corrective turn) to any one of an infinite number of possible stimuli (angles of approach). By making a number of highly oversimplifying assumptions, the problem can be reduced to a learning situation, studied previously by H. D. Landahl (Bull. Math. Biophysic,3, 13–26, 71–77, 1941). This is used not so much to obtain any relation that might be considered practically applicable immediately as toillustrate what kind of relation can be obtained from such considerations. It is shown how the safe speed of a driver depends on his total driving experience (total distance driven) as well as on his psychophysical parameters.     

Identifying the conditions for development of beneficial mycelium morphology for chitosan-producing Absidia spp. in submersed cultures

Summary The first step when producing chitosan from fungi grown submersed, i.e. the cultivation technique of some chitosan-producing Abisidia spp/, was studied. Various strains have their own specific requirements for development of morphology beneficial for a high growth rate. Regarding the species studied (A. coerulea, CBS 100.38; A. fusca, CBS 102.35; A. glauca, Dep. Microbial Ecology, Lund; A. repens; CBS 102.32), these demands are primarily those concerning agitation. Increasing the number of spores (from 1.4·10⁷ to 5.8·10⁷ per liter) of A. coerulea increased the initial growth rate, which on the other hand (due to agglomeration of pellets) levelled off at a comparatively earlier stage. Too low stirrer speed generally caused the formation of a pulp with a strong decline in growth rate. Elevated stirrer speeds caused the formation of a more compact morphology, often in the form of small pellets. The use of an open paddle impeller with a large diameter (with satisfactory pumping and mixing characteristics) ran at a sufficiently high stirrer speed to avoid the formation of a viscous pulp allowed most of the species to grow satisfactorily. Continuous cultivations were hard to establish due to lack of new growth spots and accreted growth.     

Morphology,cell growth,and polysaccharide production of <Emphasis Type="Italic">Nostoc flagelliforme</Emphasis> in liquid suspension culture at different agitation rates

Noscoc flagelliforme is a terrestrial macroscopic cyanobacterium with high economic value. Free-living cells that were separated from a natural colony of N. flagelliforme were cultivated in a 20-L photobioreactor for 16 days at five agitation rates with impeller tip speeds at 0.3, 0., 0.8, 1.0, and 1.5 m·s⁻¹. With different impeller tip speeds there were significant differences in the cell growth and polysaccharide production, and different types of cell colonies appeared because of different shear forces caused by agitation. At harvest time, cell concentrations with tip speeds of 0.8 and 1.0 m·s⁻¹ were clearly higher than those with the other three tip speeds, but dry cell weights with the tip speeds of 0.3, 0.5, 0.8, and 1.0 m·s⁻¹ were almost the same. The highest RPS (polysaccharide that released into liquid medium) production was obtained with the tip speeds of 0.8 and 1.0 m·s⁻¹, while the highest EPS (polysaccharide that formed capsule or slime layer) production was obtained with the tip speed of 0.5 m·s⁻¹. The tip speed of 1.5 m·s⁻¹ was harmful for both cell growth and polysaccharide production, indicating that an appropriate shear force was needed in the liquid suspension culture of N. flagelliforme.     

The interplay between speed,kinetics, and hand postures during primate terrestrial locomotion

Nonprimate terrestrial mammals may use digitigrade postures to help moderate distal limb joint moments and metapodial stresses that may arise during high‐speed locomotion with high‐ground reaction forces (GRF). This study evaluates the relationships between speed, GRFs, and distal forelimb kinematics in order to evaluate if primates also adopt digitigrade hand postures during terrestrial locomotion for these same reasons. Three cercopithecine monkey species (Papio anubis, Macaca mulatta, Erythrocebus patas) were videotaped moving unrestrained along a horizontal runway instrumented with a force platform. Three‐dimensional forelimb kinematics and GRFs were measured when the vertical force component reached its peak. Hand posture was measured as the angle between the metacarpal segment and the ground (MGA). As predicted, digitigrade hand postures (larger MGA) are associated with shorter GRF moment arms and lower wrist joint moments. Contrary to expectations, individuals used more palmigrade‐like (i.e. less digitigrade) hand postures (smaller MGA) when the forelimb was subjected to higher forces (at faster speeds) resulting in potentially larger wrist joint moments. Accordingly, these primates may not use their ability to alter their hand postures to reduce rising joint moments at faster speeds. Digitigrady at slow speeds may improve the mechanical advantage of antigravity muscles crossing the wrist joint. At faster speeds, greater palmigrady is likely caused by joint collapse, but this posture may be suited to distribute higher GRFs over a larger surface area to lower stresses throughout the hand. Thus, a digitigrade hand posture is not a cursorial (i.e. high speed) adaptation in primates and differs from that of other mammals. Am J Phys Anthropol 2010. © 2009 Wiley‐Liss, Inc.     

Conceptual and statistical problems with the DEC+J model of founder‐event speciation and its comparison with DEC via model selection

Phylogenetic studies of geographic range evolution are increasingly using statistical model selection methods to choose among variants of the dispersal‐extinction‐cladogenesis (DEC) model, especially between DEC and DEC+J, a variant that emphasizes “jump dispersal,” or founder‐event speciation, as a type of cladogenetic range inheritance scenario. Unfortunately, DEC+J is a poor model of founder‐event speciation, and statistical comparisons of its likelihood with DEC are inappropriate. DEC and DEC+J share a conceptual flaw: cladogenetic events of range inheritance at ancestral nodes, unlike anagenetic events of dispersal and local extinction along branches, are not modelled as being probabilistic with respect to time. Ignoring this probability factor artificially inflates the contribution of cladogenetic events to the likelihood, and leads to underestimates of anagenetic, time‐dependent range evolution. The flaw is exacerbated in DEC+J because not only is jump dispersal allowed, expanding the set of cladogenetic events, its probability relative to non‐jump events is assigned a free parameter, j, that when maximized precludes the possibility of non‐jump events at ancestral nodes. DEC+J thus parameterizes the mode of speciation, but like DEC, it does not parameterize the rate of speciation. This inconsistency has undesirable consequences, such as a greater tendency towards degenerate inferences in which the data are explained entirely by cladogenetic events (at which point branch lengths become irrelevant, with estimated anagenetic rates of 0). Inferences with DEC+J can in some cases depart dramatically from intuition, e.g. when highly unparsimonious numbers of jump dispersal events are required solely because j is maximized. Statistical comparison with DEC is inappropriate because a higher DEC+J likelihood does not reflect a more close approximation of the “true” model of range evolution, which surely must include time‐dependent processes; instead, it is simply due to more weight being allocated (via j) to jump dispersal events whose time‐dependent probabilities are ignored. In testing hypotheses about the geographic mode of speciation, jump dispersal can and should instead be modelled using existing frameworks for state‐dependent lineage diversification in continuous time, taking appropriate cautions against Type I errors associated with such methods. For simple inference of ancestral ranges on a fixed phylogeny, a DEC‐based model may be defensible if statistical model selection is not used to justify the choice, and it is understood that inferences about cladogenetic range inheritance lack any relation to time, normally a fundamental axis of evolutionary models.     

Critical current speeds and microhabitats of the benthic fishes Percina roanoka and Etheostoma flabellare

Synopsis Critical current speed (maximum current speed at which benthic stream fishes are able to hold station without active swimming) was determined in a laboratory flow chamber for Percina roanoka and Etheostoma flabellare. Mean critical current speeds differed significantly among adult P. roanoka, adult E. flabellare, and juvenile E. flabellare, and were consistent with microhabitat differences among these darters in natural environments. Behavior of E. flabellare enhanced its station-holding ability, but tests with preserved individuals suggested that morphological differences explained more of the difference in critical current speeds of the darters. On a smooth surface where grasping of the substrate was not possible, use of pectoral fins did not significantly enhance the ability of darters to maintain position in flowing water.     

Dynamics of the Terrestrial Mammals of Ipolytarnóc (Northern Hungary)

Much has been discovered about the movement speed of extant animals through the use of a variety of instruments, but much less is known about the speeds of extinct animals. The anatomy and mobility of prehistoric animals can be deduced from their bone remains and ichnofossils. By the use of biomechanics, their movement speeds also can be determined. Many ichnofossils have been found at Ipolytarnóc (the Miocene footprint locality at Ipolytarnóc, Hungary.) Ichnofossils of the most studied mammals from this site have been used in this study: the herbivores Rhinoceripeda tasnadyi, Megapecoripeda miocaenica, and Pecoripeda hamori, and the carnivore Mustelipeda punctata. The movement speed of each species was calculated from its foot and stride lengths. At Ipolytarnóc, Rhinoceripeda tasnadyi was able to move with a speed of 4.29 m/s, while Megapecoripeda miocaenica could reach a speed of 3.01 m/s. In the case of Pecoripeda hamori, its speed was 4.53 m/s, while in the case of Mustelipeda punctata it was 2.05 m/s. Conclusions about the paleoenvironment and the movement speeds of the prehistoric animals are based on our results. At times, the ancient mammals of Ipolytarnóc wallowed in puddles and muddy pools.